A systematic delayed feedback control approach for unstable periodic orbits in chaotic systems with unknown parameters

Delayed feedback control is a common technique for the stabilization of unstable periodic orbits in chaotic systems. Most results in this area rely on linear control techniques, which do not have guaranteed performance for nonlinear systems. Whereas nonlinear control methods have already been proposed, the existing results often rely on knowledge or partial knowledge of the system parameters and also have no mechanism for reaching the possible natural responses of the underlying uncontrolled system with embedded unstable periodic orbits. The main contribution of this paper is to propose a control method for a class of chaotic systems with entirely unknown parameters. Based on the delayed information of the system states, we develop an adaptive control strategy such that the system states converge to a periodic response with a desired time period. While infinitely many periodic responses are possible, the proposed control strategy is equipped with a mechanism that steers the dynamics toward the possible natural periodic responses of the system, such as unstable periodic orbits, where the controller is noninvasive. The proposed control strategy is demonstrated through simulation and used for revealing unstable periodic orbits inside the chaotic attractor of a Duffing oscillator.